1. You are given the following two sets of prices of European options as a function of the strike price, for a stock with S = 100. Assume that all options mature in 6 months,
and that the interest rate (continuously compounding, annualized) is 10%.
(1) p(90) = 4; p(100) = 9 1/8 ; p(110) = 16; p(120) = 25 3/4
(2) p(90) = 2 ¾ ; p(100) = 81/2 ; p(110) = 17; p(120) = 24
For each set of prices, please answer the following questions:
(a) Assume that the stock will not pay any dividend in the next 6 months. Do
these prices satisfy arbitrage restrictions on options values? If yes, prove it. If
not, construct an arbitrage portfolio to realize riskless pro_ts and show how that
portfolio performs whatever the underlying price does.
(b) Would your answer to Part (a) change if these put options are Americans? Why?
(c) Would your answer to Part (a) change if the stock will pay an unknown amount
of dividend in 3 months? Explain why.
To determine whether the given sets of prices satisfy arbitrage restrictions on options values, we need to check if there are any opportunities for risk-free profits.
(a) Arbitrage Restrictions:
Arbitrage restrictions on options values state that it should not be possible to create a portfolio that guarantees a risk-free profit. In our case, we have four European option prices for each set of prices.
The formula to calculate the price of a European call option is:
C = S * e^(-q*T) * N(d1) - X * e^(-r*T) * N(d2)
where:
S = stock price
q = dividend yield
T = time to expiration
r = interest rate
X = strike price
N(x) = cumulative standard normal distribution function
d1 = (ln(S/X) + (r - q + σ^2/2) * T) / (σ * sqrt(T))
d2 = d1 - σ * sqrt(T)
σ = volatility
Since dividend (q) is zero in this case, the formula simplifies to:
C = S * N(d1) - X * e^(-r*T) * N(d2)
Similarly, the formula to calculate the price of a European put option is:
P = X * e^(-r*T) * N(-d2) - S * N(-d1)
Now, let's check each set of prices:
Set 1:
p(90) = 4
p(100) = 9 1/8
p(110) = 16
p(120) = 25 3/4
For example, let's consider the p(100) option. We know that an option's price is determined by its strike price, current stock price, time to expiration, interest rate, and volatility. Given that the stock price is currently S = 100, and the interest rate is r = 10% (0.1), we can plug in these values into the option pricing formula to calculate the expected price for p(100):
p(100) = S * N(d1) - X * e^(-r*T) * N(d2)
Assuming no dividends and T = 0.5 (6 months), we can rearrange the formula to solve for X:
X = (S * N(d1) - p(100)) / (e^(-r*T) * N(d2))
Similarly, we can calculate X for other options using their respective prices.
Now, we have the strike prices for each option. We can compare these calculated strike prices with the actual given strike prices to check if there are any discrepancies. If the calculated strike price differs significantly from the given strike price, it suggests an arbitrage opportunity.
Note: In the calculations, we assume that the volatility (σ) is the same for all options for simplicity.
(b) If the put options are Americans, the answer to Part (a) would not change significantly. The European options' pricing formulas can also be applied to American options, assuming no dividends. However, since American options can be exercised at any time before expiration, they may have additional intrinsic value compared to European options.
(c) If the stock will pay an unknown amount of dividend in 3 months, it may affect the option prices as dividend payments reduce the stock price. This change in stock price will indirectly impact the strike price calculation and, consequently, the arbitrage restrictions. One would need to consider the expected dividend amount and its impact on the stock price when determining arbitrage opportunities.